Hybrid Annealing Krill Herd and Quantum-Behaved Particle Swarm Optimization

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Article Hybrid Annealing Krill Herd and Quantum-Behaved Particle Swarm Optimization

Cheng-Long Wei 1 and Gai-Ge Wang 1,2,3,4,*

1 Department of Computer Science and Technology, Ocean University of China, Qingdao 266100, China; [email protected] 2 Institute of Algorithm and Big Data Analysis, Northeast Normal University, Changchun 130117, China 3 School of Computer Science and Information Technology, Northeast Normal University, Changchun 130117, China 4 Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012, China * Correspondence: [email protected]

Received: 27 July 2020; Accepted: 17 August 2020; Published: 21 August 2020

Abstract: The particle swarm optimization algorithm (PSO) is not good at dealing with discrete optimization problems, and for the krill herd algorithm (KH), the ability of local search is relatively poor. In this paper, we optimized PSO by quantum behavior and optimized KH by simulated annealing, so a new hybrid algorithm, named the annealing krill quantum particle swarm optimization (AKQPSO) algorithm, is proposed, and is based on the annealing krill herd algorithm (AKH) and quantum particle swarm optimization algorithm (QPSO). QPSO has better performance in exploitation and AKH has better performance in exploration, so AKQPSO proposed on this basis increases the diversity of population individuals, and shows better performance in both exploitation and exploration. In addition, the quantum behavior increased the diversity of the population, and the simulated annealing strategy made the algorithm avoid falling into the local optimal value, which made the algorithm obtain better performance. The test set used in this paper is a classic 100-Digit Challenge problem, which was proposed at 2019 IEEE Congress on Evolutionary Computation (CEC 2019), and AKQPSO has achieved better performance on benchmark problems.

Keywords: swarm intelligence; simulated annealing; krill herd; particle swarm optimization; quantum

1. Introduction With the development of modern technology, artificial intelligence is becoming more and more important in society, and more and more mature, and can be used to deal with many problems that cannot be solved by traditional methods, such as wind energy decision system (WEDS) [1] and social cognitive radio network (SCRN) [2]. There are many kinds of classification methods; the simplest classification method is divided into the traditional method and modern intelligent method [3]. The traditional optimization algorithms generally deal with structured problems. The algorithms are deterministic and have only one global optimal solution. Meanwhile, the intelligent optimization algorithms generally deal with a more general description of the problems, which is heuristic, and have multiple extreme values. Intelligent optimization algorithms require certain strategies to prevent falling into the local optimum and try to find the global optimum, such as scheduling [4–7], image [8–10], feature selection [11–13] and detection [14,15], path planning [16,17], cyber-physical social system [18,19], texture discrimination [20], factor evaluation [21], saliency detection [22], classification [23,24], object extraction [25], gesture segmentation [26], economic load dispatch [27,28], shape design [29], big data and large-scale optimization [30,31], signal processing [32], multi-objective optimization [33,34], big

Mathematics 2020, 8, 1403; doi:10.3390/math8091403 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 1403 2 of 23 data optimization [30,31], unit commitment [35], vehicle routing [36], knapsack problem [37–39], fault diagnosis [40–42], and test-sheet composition [43]. Because intelligent algorithms are not strictly dependent on the mathematical relationship, and because the problems that need to be solved are becoming more and more complex, intelligent algorithms are widely used to solve various complex optimization problems. As a result, the frequency of intelligent algorithms has exceeded that of traditional algorithms. At present, some main intelligent algorithms are widely used, such as genetic algorithm (GA) [44,45], particle swarm optimization algorithm (PSO) [46–48], krill herd (KH) [49–54], ant colony optimization algorithm (ACO) [55–57], differential evolution algorithm (DE) [58,59], adaptive island evolutionary algorithm (AIE) [60], and delayed start parallel evolutionary algorithm (DSPE) [61]. The swarm intelligence algorithms mainly come from the bionic idea, which is a set of methods summarized by referring to the laws of life and predatory behavior of animals in the biological world. A swarm intelligence algorithm is a kind of algorithm that works by studying the collective behavior of animals, and the most famous swarm intelligence algorithms are particle swarm optimization algorithm (PSO) [46] and ant colony optimization algorithm (ACO) [55]. An evolutionary algorithm is mainly a kind of method obtained by studying the process of biological genetic change, and these include genetic algorithm (GA) [44], genetic programming (GP) [62], evolutionary strategy (ES) [63,64], differential evolution (DE) [58] and other algorithms are based on genes. The krill herd (KH) [49] and quantum-behaved particle swarm optimization (QPSO) [65] algorithms mentioned in this paper belong to the swarm intelligence algorithm. The KH algorithm has the advantages of simplicity, flexibility, and high computational efficiency, and the QPSO algorithm has a relatively fast convergence speed, but when they are used to solve the 100-Digit Challenge problems, they do not do well. Owing to the poor local optimization ability of QPSO, it is easy to fall into the local optimal value, so it does not solve this problem well. In order to study an algorithm with very high accuracy, as well as to simultaneously optimize the exploitation and exploration and improve the accuracy of annealing krill quantum particle swarm optimization (AKQPSO), we studied the KH algorithm with strong exploitation and the QPSO algorithm with strong exploration. By combining their advantages, the new algorithm overcomes their original shortcomings and has a strong ability in exploration and exploitation. We combine the advantages of KH and QPSO, and optimize them, forming the annealing krill quantum particle swarm optimization (AKQPSO) algorithm. The new algorithm solved the 100-Digit Challenge better. Moreover, the study solved the problem of the poor accuracy of general algorithms, and exploitation and exploration cannot achieve the optimal at the same time. In this paper, the 100-Digit Challenge problem is difficult to solve by traditional methods, and we can solve this problem better by using a swarm intelligence algorithm. In 2002, academician Nick Trefethen of Oxford University and the Society for Industrial and Applied Mathematics (SIAM) jointly developed the 100-Digit Challenge problem, which was proposed mainly to test high-precision computing [18]. The challenge consists of ten problems, each of which needs to be accurate to ten decimal places, with a total of 100 digits, so the problem is named the 100-Digit Challenge. However, traditional methods need a lot of computation to solve this challenge, and they cannot get good results, so we use swarm intelligence algorithm to solve these problems, and get satisfactory results. The rest of paper is organized as follows. Most representative studies regarding the KH and QPSO algorithm are reviewed in Section2, and these two algorithms are introduced in Section3. The proposed algorithm is described in Section4 and the experimental results of AKQPSO are presented in Section5. Section6 provides the main conclusions and highlights future work.

2. Related Work In order to illustrate the hybrid method, we introduced KH and PSO algorithms. In the following, we will introduce the KH algorithm, PSO algorithm, and 100-Digit Challenge problem, respectively. Mathematics 2020, 8, 1403 3 of 23

2.1. KH At present, we have two kinds of optimizations for the KH algorithm [49]; one is to improve the KH algorithm itself, and the other is to improve the KH by other excellent operators or algorithms. Wang et al. [54] changed the parameters of the KH algorithm and improved the speed of global convergence through chaos theory. Then, in order to improve the performance, they [66] used harmony search to replace the physical diffusion, which greatly improved the performance and efficiency. These algorithms belong to the first category, which optimized the KH algorithm itself. The following are the second category, which optimized the algorithm with better strategies. Abualigah et al. [50] put forward a new algorithm, which combined harmony search algorithm with he KH algorithm to generate a new probability factor and improved exploration search ability, so as to get a better solution. Another algorithm is proposed by Abualigah et al. [67], which combined the objective function with the KH algorithm, and had better performance in solving the problem of text clustering. Niu et al. [68] sped up the exploration convergence and improved the efficiency by using the opposite learning strategy, using a sinusoidal graph to change the inertia weight, and modifying the search process according to the inertia weight and acceleration coefficient.

2.2. PSO The PSO algorithm is one of the most famous swarm intelligence algorithms [47,69,70]. It was proposed in 1995, and it has the advantages of few parameters, a simple principle, and fast convergence. This algorithm has been widely used. For example, Sun et al. [71] used the agent model to assist the PSO algorithm to solve complex optimization problems. Through the combination of exploration and exploitation, they can better solve high-dimensional problems with limited resources. Similarly, for high-dimensional problems, Tran et al. [72] changed the fixed length of feature selection in the PSO algorithm, so that the particle swarm had a shorter length. This operation reduced the search space and gave shorter particles better performance. Thus, the overall efficiency of the PSO algorithm is improved, and it is also more suitable for high-dimensional problems. For the feature selection of the PSO algorithm, they [73] also optimized other methods. They improved the algorithm by discretizing the feature selection, and proved that the single variable discretization may reduce the performance of the feature selection stage, so they proposed a new discretization method, and got better performance. Zhang et al. [74] considered that there was no PSO algorithm that can work in noisy and noiseless environment at the same time, so they proposed the dual-environment PSO algorithm. This new algorithm is based on the top-k elite particles to search, which not only guaranteed the good performance in the noise environment, but also can be re-applied to the noise-free environment, so it filled a gap in the field of the PSO algorithm.

2.3. 100-Digit Challenge The 100-Digit Challenge problem was originally proposed by SIAM [75], which is a test method for accuracy calculation, and has been studied by many experts. Many algorithms have been applied to this challenge, and some of them are hybrid algorithms that are used to challenge this problem. Epstein et al. [76] combined the genetic algorithm (GA) with differential evolution (DE) algorithm to search the genetic space and find the latest solution. The new algorithm improved the ability to find the right answer through the fitness-based opposition and tide mutation. Brest et al. [77] used the self-adaptive differential evolution (jDE) algorithm in the DE algorithm to solve this problem, and also got better results. Different from the above algorithms, Zhang et al. [78] proposed a new DE algorithm named collective information powered DE. This new algorithm proposed a restart mechanism through collective population information to improve the robustness. Mathematics 2020, 8, 1403 4 of 23

3. KH and PSO KH and PSO are very eﬃcient algorithms in the ﬁeld of swarm intelligence, and we will respectively introduce them in the following.

3.1. KH In the process of predation, the predator will change the distribution of krill population, which will make them move rapidly, and then cause their distribution density to decrease and the distance between the predator and the food to become more and more far, which is the initial stage of KH. In this process, the distribution of krill population is determined by the following three situations: the influence of other krill individuals, behavior of getting food, and random diffusion. The KH algorithm can be described as follows: dX i = N + F + D (1) dt i i i where Ni is the influence of other krill individuals, Fi is the behavior of getting food, and Di is the behavior of random diffusion; i = 1, 2, ... , N, and N is the population size. For the influence of other krill individuals, the motion Ni,new of krill i induced by other krill is defined as follows: Ni,new = Nmaxαi + ωnNi,old (2) where Nmax represents the maximum induced velocity, Ni,old represents the previously induced movement, ωn represents the inertia weight and the value range is (0,1), and αi indicates that the individuali is affected by the induction direction of the surrounding neighbors. The next behavior Fi is to get food, as follows:

Fi = V f βi + ω f Fi,old (3)

1 where Vf is the maximum foraging speed, and its value is a constant, which is 0.02 (ms− ); ωf is the inertia weight of foraging movement, and its range is (0, 1); Fi,old is the previous foraging movement; and βi is the foraging direction. The individual Di in the last behavior can be represented as follows:

I Di = Dmax(1 )δ (4) − Imax where Dmax represents the maximum random diﬀusion speed; δ represents the direction of random diﬀusion; and I and Imax represent the current number and the maximum number of iterations, respectively. From above process, we can get the krill update process of the KH algorithm as follows:

dX X (t + ∆t) = X (t) + ∆t i (5) i i dt

XNV ∆t = Ct (UB LB ) (6) j − j j=1 where ∆t is the time interval related to the speciﬁc application; NV is the dimension of the decision variable; step factor Ct is a constant between 0 and 2; and UBj and LBj are the upper and lower bounds of corresponding variable j (j = 1, 2, ... , NV), respectively. The process of the KH algorithm (Algorithm 1) is as follows. Mathematics 2020, 8, 1403 5 of 23

Algorithm 1. KH [49] 1. Begin 2. Step 1: Initialization. Initialize the generation counter G, the population P, Vf, Dmax, and Nmax. 3. Step 2: Fitness calculation. Calculate fitness for each krill according to its initial position. 4. Step 3: While G < MaxGeneration do 5. Sort the population according to their fitness. 6. for i = 1:N (all krill) do 7. Perform the following motion calculation. 8. Motion induced by other individuals 9. Foraging motion 10. Physical diffusion 11. Implement the genetic operators. 12. Update the krill position in the search space. 13. Calculate fitness for each krill according to its new position 14. end for i 15. G = G + 1. 16. Step 4: end while. 17. End.

3.2. PSO In the PSO algorithm, each individual is called a “particle”, and each particle will ﬁnd a potential optimal solution at each iteration. Assuming that the size of the population is N, each particle i (i = 1, 2, ... , N) in the population has its own initial position Xi (Xi = xi1, xi2, ... , xid) and initial velocity Vi (Vi = vi1, vi2, ... , vid), and they will search for the optimal solution in D-dimensional space according to their own individual extremum pbest and global extremum gbest. Individual extremum is the current best point found by each particle in the search space, and global extremum is the current best point found by the whole particle group in the search space. During the search process, the updating formula of particle’s relevant state parameters is as follows:

vt+1 = ηvt + c rand1() (pt xt) + c rand2() (gt xt) (7) i i 1 ∗ · best − i 2 ∗ · best − i t+1 = t + t xi xi vi (8) t η = ηstart (ηstart η ) (9) − − end T where η is the inertia weight that determines the specific gravity of the particle to the current velocity. If the η is large, the particle has a strong exploration ability and can span a longer distance to find the global optimal solution, but it may cause the particle to oscillate back and forth before and after the optimal solution. If the η is small, it means that particles have better ability to find the optimal solution locally, but they can easily to fall into the local optimization solution. The flow of the standard PSO algorithm (Algorithm 2) is given below, where bup and blo represent the upper and lower bounds of the problem domain, respectively, and D represents the dimension of the solution space. Mathematics 2020, 8, 1403 6 of 23

Algorithm 2. PSO [46] Begin Step 1: Initialization. 1. Initial position: the initial position of each particle obeys uniform distribution, that is X U(bup, b ). i ∼ lo 2. Initialize its own optimal solution and the overall optimal solution: the initial position is its own optimal solution pi = xi, and then calculating the corresponding value of each particle according to the defined utility function f, and find the global optimal solution gbest. 3. Initial speed: the speed also obeys the uniform distribution. Step 2: Update. According to Equations (7) and (8), the velocity and position of particles are updated, and the current fitness of particles is calculated according to the utility function f of the problem. If it is better than its own historical optimal solution, it will update its own historical optimal solution pbest, otherwise it will not update. If the particle’s own optimal solution pbest is better than the global optimal solution g, then the global optimal solution g is updated, otherwise it is not updated. Step 3: Determine whether to terminate: Determine whether the best solution meets the termination conditions, if yes, stop. Otherwise, return to Step 2. End.

As shown in Equations (7) and (8), during each iteration, the particles update the direction and speed of the next ﬂight based on their own and group experience. The main characteristics of the PSO algorithm are as follows:

1. Particles have memory. Each iteration of particles will transfer the optimal solution of the population to each other, and update the database of all particles. If the particles deviate, the direction and velocity can be corrected by self-cognition and group-cognition. 2. PSO algorithm has fewer parameters, so it is easy to adjust, and the structure is simple, so it is easy to implement. 3. The operation is simple, and it only searches for the optimal solution in the solution space according to the ﬂight of particles.

4. AKQPSO The AKQPSO algorithm is a mixed metaheuristic algorithm, which combines the advantages of annealing krill herd (AKH) and QPSO [79]. AKH solves the disadvantage that KH cannot escape from the local optimal solution. At the same time, the eﬃciency of KH algorithm is improved by the simulated annealing strategy. Quantum behaved PSO (QPSO) is a new algorithm proposed by Sun et al. [79] in 2004. By introducing the quantum behavior and combining it with the idea of simulated annealing, the search ability of AKQPSO is greatly improved, and the new algorithm has better performance. The principle of QPSO is shown below. Quantum computing is a new computing mode, which follows the laws of quantum mechanics and regulates the quantum information unit. In quantum space, when the aggregation state property is satisﬁed, particles can search in the whole feasible solution space, thus greatly improving the exploration search ability of QPSO. According to the analysis theory of particle convergence trajectory, if every particle can converge to its local attraction point Pi = (pi1, pi2 ... , pid), then the algorithm has the possibility of convergence. The particle position update expression of the standard QPSO algorithm is as follows:

xt+1 = pt α Ct xt ln[1/ut ], ut U(0, 1) (10) ij ij ± · ij − ij · ij ij ∼ where α is the only parameter that needs to be adjusted in the algorithm, called the compression-expansion factor, which is used to control the convergence rate of particles. During the iterative process, the calculation method of individual and global optimal position is the same Mathematics 2020, 8, 1403 7 of 23 as that of the PSO algorithm, and the biggest diﬀerence is that the QPSO algorithm removes the speed information. First of all, we initialize the whole population, and all the individuals in the population are randomly generated. After initialization, we divide the population into two subpopulations according to the ratio of 3:7, which is discussed in Section 5.2. The population with the proportion of 3/10 is optimized by the improved KH algorithm, which is called subpopulation-AKH, and the population with the proportion of 7/10 is optimized by the QPSO algorithm, which is called subpopulation-QPSO. The two subpopulations will be re-integrated into a population after iterative optimization. If the optimization result of the new population meets the output conditions, then the result can be output. However, if it does not meet the output conditions, then it can be re-divided according to the proportion of 3:7, and repeat the above process until the results meet the termination conditions. In the process of decentralized and re-fusion of this population, the location information of individuals is shared, so it will greatly improve the eﬃciency of fusion search and get the best results faster. In the process of the AKQPSO algorithm, we also optimized the KH algorithm by the idea of simulated annealing, named annealing KH (AKH), so we used QPSO and simulated annealing strategy to guarantee the algorithm to escape from the local optimal value. The framework of AKQPSO is shown in Figure1 and the process of the AKQPSO algorithm (Algorithm 3) is as follows.

Algorithm 3 AKQPSO Initialization: N random individuals were generated. Set initialization parameters of AKH and QPSO. Evaluation: Evaluate all individuals based on location. Partition: The whole population was divided into subpopulation-AKH and subpopulation-QPSO. AKH process: Subpopulation-AKH individuals were optimized by AKH. Update through these three actions of the influence of other krill individuals, behavior of getting food and random diffusion. The simulated annealing strategy is used to deal with the above behaviors. Update the individual position according to the above behavior. QPSO process: Subpopulation-QPSO individuals were optimized by QPSO. Update the particle’s local best point Pi and global best point Pbest. t+1 Update the position xij by Equation (10). Combination: The population optimized by AKH and QPSO was reconstituted into a new population. Finding the best solution: The fitness of all individuals was calculated, and the best solution was found in the newly-combined population. Determine whether to terminate: Determine whether the best solution meets the termination conditions, if yes, stop. Otherwise, return to step Evaluation. Mathematics 2020, 8, x FOR PEER REVIEW 7 of 25

tt+1 =±⋅−⋅α tt tt xpij ij Cx ij ijln[1/ uuU ij ], ij ~ (0,1) (10) where α is the only parameter that needs to be adjusted in the algorithm, called the compression-expansion factor, which is used to control the convergence rate of particles. During the iterative process, the calculation method of individual and global optimal position is the same as that of the PSO algorithm, and the biggest difference is that the QPSO algorithm removes the speed information. First of all, we initialize the whole population, and all the individuals in the population are randomly generated. After initialization, we divide the population into two subpopulations according to the ratio of 3:7, which is discussed in Section 5.2. The population with the proportion of 3/10 is optimized by the improved KH algorithm, which is called subpopulation-AKH, and the population with the proportion of 7/10 is optimized by the QPSO algorithm, which is called subpopulation-QPSO. The two subpopulations will be re-integrated into a population after iterative optimization. If the optimization result of the new population meets the output conditions, then the result can be output. However, if it does not meet the output conditions, then it can be re-divided according to the proportion of 3:7, and repeat the above process until the results meet the termination conditions. In the process of decentralized and re-fusion of this population, the location information of individuals is shared, so it will greatly improve the efficiency of fusion search and get the best results faster. In the process of the AKQPSO algorithm, we also optimized the KH algorithm by the idea of simulated annealing, named annealing KH (AKH), so we used QPSO and simulated annealing strategy to guarantee the algorithm to escape from the local optimal value. The framework of AKQPSO is shown in Figure 1 and the process of the AKQPSO algorithm (Algorithm

3) is as follows.Mathematics 2020, 8, 1403 8 of 23

Start

Initialize the AKQPSO

Evaluate the individuals in the population

Partition the population

Motion induced by other individuals

Calculate p(i) and g(best) Foraging motion

AKH QPSO

Physical diffusion Up date the posi tion x(i) by Equation (11)

Simulated annealing strategy

Combine the two subpopulations into one population

Find and record the best solution at present.

N Whether the termination conditions are met

Output the optimal result

End Figure 1. Framework of annealing krill quantum particle swarm optimization (AKQPSO). Figure 1. Framework of annealing krill quantum particle swarm optimization (AKQPSO). 5. Simulation Results The 100-Digit Challenge problem comes from CEC 2019 and can be found through the website http://www.ntu.edu.sg/home/EPNSugan/index_files/CEC2019. This challenge includes 10 problems, which are 10 functions in our data. The goal is to calculate the accuracy of the function to 10 digits without a time limit. The proposed algorithm was written in MATLAB R2016a and was run in the following conditions: Intel(R) Core(TM) i5-8100 CPU 3.60 GHz, which possesses 8G of RAM on Windows 10. Table1 shows the basic parameters of the 100-Digit Challenge. The 10 test problems are the definition of minimization and some other definitions are as follows:

T Min f (x), x = [x1, x2, ... , xD] (11)

where D is the dimension, the global optimal value of the shift is randomly distributed in ( 80, 80), − and all testing problems are scalable.

Table 1. The basic parameters of the 100-Digit Challenge.

* ( *) No. Functions Fi =Fi x D Search Range 1 Storn’s Chebyshev Polynomial Fitting Problem 1 9 ( 8192–8192) − 2 Inverse Hilbert Matrix Problem 1 16 ( 16,384–16,384) − 3 Lennard–Jones Minimum Energy Cluster 1 18 ( 4–4) − 4 Rastrigin’s Function 1 10 ( 100–100) − 5 Griewangk’s Function 1 10 ( 100–100) − 6 Weierstrass Function 1 10 ( 100–100) − 7 Modiﬁed Schwefel’s Function 1 10 ( 100–100) − 8 Expanded Schaﬀer’s F6 Function 1 10 ( 100–100) − 9 Happy Cat Function 1 10 ( 100–100) − 10 Ackley Function 1 10 ( 100–100) − Mathematics 2020, 8, 1403 9 of 23

5.1. The Comparison of AKQPSO and Other Algorithms In this paper, we compared the AKQPSO algorithm with the following eight state-of-the-art algorithms.

1. By improving krill migration operator, biogeography-based KH (BBKH) [51] was proposed. 2. By adding a new hybrid differential evolution operator, the efficiency of the updating process is improved, and differential evolution KH (DEKH) [80] was proposed. 3. By quantum behavior to optimize KH, quantum-behaved KH (QKH) [65] was proposed. 4. By adding the stud selection and crossover operator, the efficiency was improved and stud krill herd (SKH) [81] was proposed. 5. By adding chaos map to optimize cuckoo search (CS), the chaotic CS (CCS) [82] was proposed. 6. By adding variable neighborhood (VN) search to bat algorithm (BA), VNBA [83] was proposed. 7. By discussing physical biogeography and its mathematics, biogeography-based optimization (BBO) [84] was proposed. 8. Genetic algorithm (GA) [45] is basic algorithm of evolutionary computing.

Table2 is the common parameter setting of these algorithms. We set the population size ( N) to 50; after 7500 iterations (max_gen) and 100 independent runs (max_run), all algorithm experiments are carried out under the condition that the basic parameters are consistent.

Table 2. Parameter settings.

Parameters N max_gen max_run Value 50 7500 100

Table3 shows the optimization results of AKQPSO and eight other algorithms for ten of the 100-Digits Challenge, with the best value in bold. From the experimental results, we can see that the optimal values of the ten problems are all calculated by the AKQPSO algorithm. AKQPSO is the best algorithm among these algorithms, and the result is the best in every problem, but QKH is not the second in every problem, so we can know that our algorithm improvement is very eﬀective. Through Figure2, we can clearly conclude that the performance of AKQPSO is the best in general, followed by QKH, and the performance of GA is the worst. In general, we rank the algorithms as follows: AKQPSO >MathematicsQKH > CCS2020, 8>, xBBKH FOR PEER> DEKH REVIEW> SKH > VNBA > GA. 11 of 25

Figure 2. Accuracy of AKQPSO and other algorithms on all problems. Figure 2. Accuracy of AKQPSO and other algorithms on all problems.

Figure 3. Accuracy of AKQPSO and other algorithms on problem 1: Storn’s Chebyshev polynomial fitting problem. BBKH, biogeography-based krill herd; DEKH, differential evolution KH; QKH, quantum-behaved KH; SKH, stud KH; CCS, chaotic cuckoo search; VNBA, variable neighborhood bat algorithm; BBO, biogeography-based optimization; GA, genetic algorithm.

Mathematics 2020, 8, 1403 10 of 23

Table 3. The minimum value of annealing krill quantum particle swarm optimization (AKQPSO) and other algorithms in the 100-Digit Challenge. BBKH, biogeography-based krill herd; DEKH, diﬀerential evolution KH; QKH, quantum-behaved KH; SKH, stud KH; CCS, chaotic cuckoo search; VNBA, variable neighborhood bat algorithm; BBO, biogeography-based optimization; GA, genetic algorithm. (The table results retain ten decimal places.).

Algorithm Problem 1 Problem 2 Problem 3 AKQPSO 10,893.3861385383 215.6832766302 1.3068652046 (Algorithm 3) BBKH 291,803.2852933580 613.7135086878 5.9027057769 DEKH 975,339.7308122220 708.0601338156 6.5872946609 QKH 47,583.9014728552 404.4416682185 2.1853645745 SKH 1,773,168.4809322700 402.7243166556 8.0692567398 CCS 211,411.5538712060 358.9175375272 12.4818284785 VNBA 2,550,944.5550316900 816.8071391350 4.6163144446 BBO 72,287.9373622652 311.8124377894 2.9209583205 GA 8,009,879.0206872900 2209.1904660292 9.7547945042 Algorithm Problem 4 Problem 5 Problem 6 AKQPSO 4.0398322695 1.0434696313 1.1185826442 (Algorithm 3) BBKH 17.7900443880 1.8852367864 3.9551854227 DEKH 29.4088567232 1.9507021960 3.1953451545 QKH 7.1133424631 2.0695362381 3.1213610789 SKH 50.9548872356 1.9350825919 10.2543233489 CCS 61.4604530135 3.4583111771 6.6010211976 VNBA 24.7278199397 3.0251295691 5.7134938818 BBO 17.4206259198 1.0971964590 2.5154932374 GA 53.9444360765 2.7126573212 6.2420012291 Algorithm Problem 7 Problem 8 Problem 9 AKQPSO 121.8932375270 2.3131324015 1.0715438957 (Algorithm 3) BBKH 779.4066229661 4.3846591727 1.3554069744 DEKH 1060.0387128932 4.4509969471 1.3614909209 QKH 195.0770363640 2.5711387868 1.2989726797 SKH 1348.8213407547 4.8795287164 1.5980935412 CCS 2247.4406602539 5.4002626350 1.6453371395 VNBA 644.6334455535 4.1477932707 1.3950038682 BBO 950.8018632939 3.5941524656 1.2098037868 GA 1276.4565372145 4.2393160129 1.3580083289 Algorithm Problem 10 -- AKQPSO 21.0237707978 -- (Algorithm 3) BBKH 21.6420368754 - - DEKH 21.6383024320 - - QKH 21.1582649016 - - SKH 21.6017549654 - - CCS 21.9603961872 - - VNBA 21.0799825759 - - BBO 21.9988888954 - - GA 21.4392031521 - -

Figure 2. Accuracy of AKQPSO and other algorithms on all problems.

Figures 3–12 show the results of the nine algorithms on each problem, and the experimental results show that AKQPSO algorithm has obvious advantages over other algorithms. From these column charts, we can see that the fluctuation of problem 1 is the largest. In problem 1, the results of the GAMathematics algorithm2020, 8, 1403with the worst performance are 735 times different from those of 11AKQPSO of 23 algorithm with the best performance, which can be said to be very different. Because the 100-Digits Challenge problem is used to test the computational accuracy, we can know that the computational is used to test the computational accuracy, we can know that the computational accuracy of AKQPSO accuracyalgorithm of AKQPSO is the highest algorithm among theis the nine highest algorithms. amon Ing Figure the nine3, because algorithms. the results In ofFigure AKQPSO, 3, because QKH, the resultsand of BBO AKQPSO, are so di ﬀQKH,erent from and those BBO of are GA, so the differe columnsnt offrom these those algorithms of GA, are the almost columns invisible, of sothese algorithmsthey are are not obviousalmost ininvisible, the same so chart. they However, are not in obvious fact, the improvementin the same ofchart. AKQPSO However, compared in withfact, the improvementQKH and BBOof AKQPSO is signiﬁcant. compared with QKH and BBO is significant.

FigureFigure 3. Accuracy 3. Accuracy of AKQPSO of AKQPSO and and other other algorithms algorithms ononproblem problem 1: 1: Storn’s Storn’s Chebyshev Chebyshev polynomial polynomial fittingﬁtting problem. problem. BBKH, BBKH, biogeography-based biogeography-based krill krill herd; herd; DEKH, didifferentialﬀerential evolution evolution KH; KH; QKH, QKH, quantum-behaved KH; SKH, stud KH; CCS, chaotic cuckoo search; VNBA, variable neighborhood bat Mathematicsquantum-behaved 2020, 8, x FOR KH;PEER SKH, REVIEW stud KH; CCS, chaotic cuckoo search; VNBA, variable neighborhood bat12 of 25 algorithm; BBO, biogeography-based optimization; GA, genetic algorithm. algorithm; BBO, biogeography-based optimization; GA, genetic algorithm.

FigureFigure 4. Accuracy 4. Accuracy of AKQPSO of AKQPSO and and other other algorithms algorithms on problemproblem 2: 2: inverse inverse Hilbert Hilbert matrix matrix problem. problem.

Figure 5. Accuracy of AKQPSO and other algorithms on problem 3: Lennard–Jones minimum energy cluster.

Figure 6. Accuracy of AKQPSO and other algorithms on problem 4: Rastrigin’s function.

Mathematics 2020, 8, x FOR PEER REVIEW 12 of 25 Mathematics 2020, 8, x FOR PEER REVIEW 12 of 25

Mathematics 2020, 8, 1403 12 of 23 Figure 4. Accuracy of AKQPSO and other algorithms on problem 2: inverse Hilbert matrix problem. Figure 4. Accuracy of AKQPSO and other algorithms on problem 2: inverse Hilbert matrix problem.

Figure 5. Accuracy of AKQPSO and other algorithms on problem 3: Lennard–Jones minimum energy Figure 5. 5. AccuracyAccuracy of ofAKQPSO AKQPSO and and other other algorithms algorithms on problem on problem 3: Lennard–Jones 3: Lennard–Jones minimum minimum energy cluster. cluster.energy cluster.

Mathematics 2020, 8, x FOR PEER REVIEW 13 of 25 Figure 6. Accuracy of AKQPSO and other algorithms on problem 4: Rastrigin’s function. Figure 6. Accuracy of AKQPSO and other algorithms on problem 4: Rastrigin’s function.

Figure 7. Accuracy of AKQPSO and other algorithms on problem 5: Griewangk’s function. Figure 7. Accuracy of AKQPSO and other algorithms on problem 5: Griewangk’s function.

Figure 8. Accuracy of AKQPSO and other algorithms on problem 6: Weierstrass function.

Figure 9. Accuracy of AKQPSO and other algorithms on problem 7: modified Schwefel’s function.

Mathematics 2020, 8, x FOR PEER REVIEW 13 of 25 Mathematics 2020, 8, x FOR PEER REVIEW 13 of 25

Figure 7. Accuracy of AKQPSO and other algorithms on problem 5: Griewangk’s function. Mathematics 2020, 8, 1403 13 of 23 Figure 7. Accuracy of AKQPSO and other algorithms on problem 5: Griewangk’s function.

Figure 8. Accuracy of AKQPSO and other algorithms on problem 6: Weierstrass function. Figure 8. Accuracy of AKQPSO and other algorithms on problem 6: Weierstrass function. Figure 8. Accuracy of AKQPSO and other algorithms on problem 6: Weierstrass function.

Mathematics 2020, 8, x FOR PEER REVIEW 14 of 25 Figure 9. Accuracy of AKQPSO and other algorithms on problem 7: modiﬁed Schwefel’s function. Figure 9. Accuracy of AKQPSO and other algorithms on problem 7: modified Schwefel’s function. Figure 9. Accuracy of AKQPSO and other algorithms on problem 7: modified Schwefel’s function.

Figure 10. Accuracy of AKQPSO and other algorithms on problem 8: expanded Schaﬀer’s F6 function. Figure 10. Accuracy of AKQPSO and other algorithms on problem 8: expanded Schaffer’s F6 function.

Figure 11. Accuracy of AKQPSO and other algorithms on problem 9: happy cat function.

Figure 12. Accuracy of AKQPSO and other algorithms on problem 10: Ackley function.

Mathematics 2020, 8, x FOR PEER REVIEW 14 of 25

Figure 10. Accuracy of AKQPSO and other algorithms on problem 8: expanded Schaffer’s F6 MathematicsFigure2020 10., 8 ,Accuracy 1403 of AKQPSO and other algorithms on problem 8: expanded Schaffer’s F614 of 23 function.

Figure 11. Accuracy of AKQPSO and other algorithmsalgorithms on problem 9: happy cat function.

Figure 12. Accuracy of AKQPSO and other algorithms on problem 10: Ackley function. Figure 12. Accuracy of AKQPSO and other algorithms on problem 10: Ackley function. Among these ten problems, we divide the problems into four groups according to the ability of these problems to test the computational accuracy of AKQPSO. Some groups have a strong ability to test, so it is more obvious which algorithm is better. The first group: problem 1. According to the computational accuracy, the results displayed in this group are quite different. The second group: problems 2, 3, and 7. This group has higher requirements for the accuracy, so it is difficult to calculate the best value of the function. The third group: problems 4, 6, 8, and 10. This group has a slightly lower requirement for the computational accuracy, so the results of various algorithms are very close, but it is difficult to achieve the best results. The fourth group: problems 5 and 9. This group has the lowest requirements for the computational accuracy, so many algorithms can be very close to the optimal result 1.0000000000. Figures 13–16 show the advantage of AKQPSO over other algorithms in each group of the 100-Digits Challenge problem. Although the accuracy of GA is the worst when ranking from the whole, and in the first of the four groups, it is also the GA. In the other three groups, however, the algorithm with the worst accuracy is CCS algorithm, so in terms of grouping comparison, CCS algorithm is the worst. In addition, the accuracy of AKQPSO algorithm is still the best among the four groups, which has not changed. From the above analysis, AKQPSO is the best in all aspects, and the computational accuracy of this algorithm is also the highest. Mathematics 2020, 8, x FOR PEER REVIEW 15 of 25 Mathematics 2020, 8, x FOR PEER REVIEW 15 of 25 Among these ten problems, we divide the problems into four groups according to the ability of Among these ten problems, we divide the problems into four groups according to the ability of these problems to test the computational accuracy of AKQPSO. Some groups have a strong ability to these problems to test the computational accuracy of AKQPSO. Some groups have a strong ability to test, so it is more obvious which algorithm is better. The first group: problem 1. According to the test, so it is more obvious which algorithm is better. The first group: problem 1. According to the computational accuracy, the results displayed in this group are quite different. The second group: computational accuracy, the results displayed in this group are quite different. The second group: problems 2, 3, and 7. This group has higher requirements for the accuracy, so it is difficult to problems 2, 3, and 7. This group has higher requirements for the accuracy, so it is difficult to calculate the best value of the function. The third group: problems 4, 6, 8, and 10. This group has a calculate the best value of the function. The third group: problems 4, 6, 8, and 10. This group has a slightly lower requirement for the computational accuracy, so the results of various algorithms are slightly lower requirement for the computational accuracy, so the results of various algorithms are very close, but it is difficult to achieve the best results. The fourth group: problems 5 and 9. This very close, but it is difficult to achieve the best results. The fourth group: problems 5 and 9. This group has the lowest requirements for the computational accuracy, so many algorithms can be very group has the lowest requirements for the computational accuracy, so many algorithms can be very close to the optimal result 1.0000000000. close to the optimal result 1.0000000000. Figures 13–16 show the advantage of AKQPSO over other algorithms in each group of the Figures 13–16 show the advantage of AKQPSO over other algorithms in each group of the 100-Digits Challenge problem. Although the accuracy of GA is the worst when ranking from the 100-Digits Challenge problem. Although the accuracy of GA is the worst when ranking from the whole, and in the first of the four groups, it is also the GA. In the other three groups, however, the whole, and in the first of the four groups, it is also the GA. In the other three groups, however, the algorithm with the worst accuracy is CCS algorithm, so in terms of grouping comparison, CCS algorithm with the worst accuracy is CCS algorithm, so in terms of grouping comparison, CCS algorithm is the worst. In addition, the accuracy of AKQPSO algorithm is still the best among the algorithm is the worst. In addition, the accuracy of AKQPSO algorithm is still the best among the four groups, which has not changed. From the above analysis, AKQPSO is the best in all aspects, and four groups, which has not changed. From the above analysis, AKQPSO is the best in all aspects, and Mathematicsthe computational2020, 8, 1403 accuracy of this algorithm is also the highest. 15 of 23 the computational accuracy of this algorithm is also the highest.

Figure 13. Ratio of AKQPSO over other algorithms in the first group. Figure 13. RatioRatio of of AKQPSO AKQPSO over other al algorithmsgorithms in the first ﬁrst group.

Mathematics 2020, 8, x FOR PEER REVIEW 16 of 25 Figure 14. Ratio of AKQPSO over other algorithms in the second group. Figure 14. Ratio of AKQPSO over other algorithms in the second group.

Figure 15. Ratio of AKQPSO over other algorithms in the third group. Figure 15. Ratio of AKQPSO over other algorithms in the third group.

Figure 16. Ratio of AKQPSO over other algorithms in the fourth group.

5.2. Evaluation Parameter λ In Section 4, after the initialization of the population, AKQPSO algorithm divided the population into two subpopulations: AKH and QPSO. We set the parameter λ, which represents the proportion of the two subpopulations. In the experiment, λ = 0.1, 0.2, …, 0.9. For example, if λ = 0.1, this means that the ratio of subpopulation-AKH/subpopulation-QPSO is 1:9, and other numbers mean the same thing. After a lot of experiments, the optimal parameter value is λ = 0.3. All the above experiments are based on the results obtained using λ = 0.3, and the following part will introduce the experiments on λ = 0.3. The parameters of this part of experiments are still the same as Table 2 to ensure that all experiments are tested under the same conditions. Table 4 shows the experimental results of 10 problems that are calculated by different λ in the AKQPSO algorithm in the 100-Digit Challenge. Bold font indicates the best value. From Table 4, we can see that, in the case of λ = 0.3, eight of the ten problems can get the best value. In the remaining two problems, even if λ = 0.3 does not reach the best value, its result is the second among the nine values, and it is very close to the best value. Therefore, when λ = 0.3, the performance of AKQPSO can reach the best. As can be seen from Figure 17a, the fluctuation range of the result of problem 1 is the largest. As problems 1–10 are in the same chart, and the fluctuation of problem 1 is much larger than that of

Mathematics 2020, 8, x FOR PEER REVIEW 16 of 25

Mathematics 2020, 8, 1403 16 of 23 Figure 15. Ratio of AKQPSO over other algorithms in the third group.

Figure 16. RatioRatio of AKQPSO over other algo algorithmsrithms in the fourth group. 5.2. Evaluation Parameter λ 5.2. Evaluation Parameter λ In Section4, after the initialization of the population, AKQPSO algorithm divided the population In Section 4, after the initialization of the population, AKQPSO algorithm divided the into two subpopulations: AKH and QPSO. We set the parameter λ, which represents the proportion of population into two subpopulations: AKH and QPSO. We set the parameter λ, which represents the the two subpopulations. In the experiment, λ = 0.1, 0.2, ... , 0.9. For example, if λ = 0.1, this means proportion of the two subpopulations. In the experiment, λ = 0.1, 0.2, …, 0.9. For example, if λ = 0.1, that the ratio of subpopulation-AKH/subpopulation-QPSO is 1:9, and other numbers mean the same this means that the ratio of subpopulation-AKH/subpopulation-QPSO is 1:9, and other numbers thing. After a lot of experiments, the optimal parameter value is λ = 0.3. All the above experiments mean the same thing. After a lot of experiments, the optimal parameter value is λ = 0.3. All the above are based on the results obtained using λ = 0.3, and the following part will introduce the experiments experiments are based on the results obtained using λ = 0.3, and the following part will introduce the on λ = 0.3. The parameters of this part of experiments are still the same as Table2 to ensure that all experiments on λ = 0.3. The parameters of this part of experiments are still the same as Table 2 to experiments are tested under the same conditions. ensure that all experiments are tested under the same conditions. Table4 shows the experimental results of 10 problems that are calculated by di ﬀerent λ in the Table 4 shows the experimental results of 10 problems that are calculated by different λ in the AKQPSO algorithm in the 100-Digit Challenge. Bold font indicates the best value. From Table4, we AKQPSO algorithm in the 100-Digit Challenge. Bold font indicates the best value. From Table 4, we can see that, in the case of λ = 0.3, eight of the ten problems can get the best value. In the remaining can see that, in the case of λ = 0.3, eight of the ten problems can get the best value. In the remaining two problems, even if λ = 0.3 does not reach the best value, its result is the second among the nine two problems, even if λ = 0.3 does not reach the best value, its result is the second among the nine values, and it is very close to the best value. Therefore, when λ = 0.3, the performance of AKQPSO can values, and it is very close to the best value. Therefore, when λ = 0.3, the performance of AKQPSO reach the best. can reach the best. As can be seen fromTable Figure 4. The 17a, diﬀerent the fluctuation subpopulations range in theof the AKQPSO result algorithm.of problem 1 is the largest. As problems 1–10 are in the same chart, and the fluctuation of problem 1 is much larger than that of λ Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 0.1 49,649.1118923043 371.5625700159 1.4091799599 6.9697543426 1.1082189445 0.2 133,608.1106469210 370.0473039979 1.4091358439 3.9848771713 1.0983396055 0.3 10,893.3861385383 215.6832766302 1.3068652046 4.0398322695 1.0434696313 0.4 53,539.2035950009 346.1060025716 1.4091347288 4.9798362284 1.0588561989 0.5 86,346.3589566716 424.4795442904 1.4091497973 5.9747952855 1.0564120753 0.6 82,979.4098834243 422.2483723507 1.4094790359 5.0457481023 1.0861547616 0.7 99,948.7393233432 609.3844384539 7.7057897580 7.9647083618 1.0885926731 0.8 172,794.7224327620 462.1482898652 1.4091546130 13.9344627044 1.0689273036 0.9 29,013.1541012323 412.4776865757 3.9404220234 8.9597299262 1.0642761100 λ Problem 6 Problem 7 Problem 8 Problem 9 Problem 10 0.1 1.4704093392 401.1855067018 3.0118025092 1.0958187632 21.1048204811 0.2 1.2289194470 336.4312675492 2.7987816724 1.0968167661 21.0708288643 0.3 1.1185826442 121.8932375270 2.3131324015 1.0715438957 21.0237707978 0.4 2.6943320338 119.5337169812 2.3299026264 1.1010758378 21.0389405345 0.5 2.9771884111 209.5578114314 2.5217968033 1.1257275250 21.0587530353 0.6 2.5706601162 253.1211550958 3.5227629532 1.1353127271 21.0976573470 0.7 1.4737157145 187.5895821815 3.0884949525 1.1344605652 21.1144506567 0.8 1.5487741806 144.3502633617 3.1612574448 1.1553497252 21.0835644844 0.9 4.0000000000 131.2707035329 4.0209018951 1.1615503990 21.0768903380 Mathematics 2020, 8, x FOR PEER REVIEW 18 of 25

Table 4. The different subpopulations in the AKQPSO algorithm.

λ Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 0.1 49,649.1118923043 371.5625700159 1.4091799599 6.9697543426 1.1082189445 0.2 133,608.1106469210 370.0473039979 1.4091358439 3.9848771713 1.0983396055 0.3 10,893.3861385383 215.6832766302 1.3068652046 4.0398322695 1.0434696313 0.4 53,539.2035950009 346.1060025716 1.4091347288 4.9798362284 1.0588561989 0.5 86,346.3589566716 424.4795442904 1.4091497973 5.9747952855 1.0564120753 0.6 82,979.4098834243 422.2483723507 1.4094790359 5.0457481023 1.0861547616 0.7 99,948.7393233432 609.3844384539 7.7057897580 7.9647083618 1.0885926731 0.8 172,794.7224327620 462.1482898652 1.4091546130 13.9344627044 1.0689273036 0.9 29,013.1541012323 412.4776865757 3.9404220234 8.9597299262 1.0642761100 Mathematics 2020, 8, 1403 17 of 23 λ Problem 6 Problem 7 Problem 8 Problem 9 Problem 10 0.1 1.4704093392 401.1855067018 3.0118025092 1.0958187632 21.1048204811 As can0.2 be seen1.2289194470 from Figure 336.431267549217a, the fluctuation 2.7987816724 range of the1.0968167661 result of problem21.0708288643 1 is the largest. As problems0.3 1–101.1185826442 are in the same121.8932375270 chart, and the 2.3131324015 fluctuation of1.0715438957 problem 1 is much21.0237707978 larger than that of other0.4 problems, 2.6943320338 the fluctuation 119.5337169812 range of problems 2.3299026264 2–10 is not1.1010758378 obvious in Figure21.0389405345 17a. In order to compare0.5 problems 2.9771884111 2–10 more clearly, 209.5578114314 we used Figure2.5217968033 17b to show1.1257275250 the fluctuation 21.0587530353 of other problems 2–10. Through0.6 2.5706601162 Figure 17b, we can 253.1211550958 see that the 3.5227629532 fluctuation of1.1353127271 problems 2 and21.0976573470 7 are also obvious. Therefore,0.7 among1.4737157145 these ten problems, 187.5895821815 problems 3.0884949525 1, 2, and 7 are1.1344605652 the three problems 21.1144506567 with the largest fluctuation.0.8 Among1.5487741806 these ten problems, 144.3502633617 the fluctuation 3.1612574448 of problem 1.1553497252 7 is different 21.0835644844 from that of the other nine problems.0.9 With4.0000000000 the increasing 131.2707035329 proportion of4.0209018951 AKH and QPSO, 1.1615503990 the result of21.0768903380 problem 7 tends to be better. For other problems, however, with the increasing proportion of AKH and QPSO, their results tend to be worse, which is the opposite of problem 7.

Mathematics 2020, 8, x FOR PEER REVIEW 19 of 25 (a)

(b)

Figure 17. The different different subpopulations in didifferentfferent problems.problems. ( a)) The different different subpopulations in problems 1–10 and (b)) thethe didifferentfferent subpopulationssubpopulations inin problemsproblems 2–10.2–10.

5.3. Complexity Analysis of AKQPSO The main computing overhead of AKQPSO associated with Step “Partition” of the AKQPSO algorithm in Section 4. The following is a detailed analysis of the single computational complexity of Step “Partition” and other step in AKQPSO. N is the number of individuals in the population. 1. Step “Partition”: This step accounts for the main computational overhead, so we focus on this step. • Step “AKH process”: The computational complexity of this step mainly includes “Subpopulation-AKH individuals were optimized by AKH”, “Update through these three actions of the influence of other krill individuals, behavior of getting food and random diffusion”, “The simulated annealing strategy is used to deal with the above behaviors”, and “Update the individual position according to the above behavior”, and their complexities are O(N), O(N2), O(N), and O(N), respectively. • Step “QPSO process”: The computational complexity of this step mainly includes “Subpopulation-QPSO individuals were optimized by QPSO”, “Update the particle’s local t+1 i best best point P and global best point P ”, and “Update the position xij by Equation (10)”, and their complexities are all O(N). 2. Other step. • The computational complexity of Step “Initialization”, Step “Evaluation”, Step “Combination”, and Step “Finding the best solution” are all O(N).

Therefore, in one generation AKQPSO, the worst-case complexity is O(N2).

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From the results of Sections 5.1 and 5.2, problems 1, 2, and 7 belong to the ﬁrst and the second of the four groups, respectively, and they are also the two groups with the highest requirements for the computational accuracy of AKQPSO. Therefore, the adjustment of subpopulation has a very direct eﬀect on the accuracy of AKQPSO.

1. Step “Partition”: This step accounts for the main computational overhead, so we focus on this step.

Step “AKH process”: The computational complexity of this step mainly includes • “Subpopulation-AKH individuals were optimized by AKH”, “Update through these three actions of the inﬂuence of other krill individuals, behavior of getting food and random diﬀusion”, “The simulated annealing strategy is used to deal with the above behaviors”, and “Update the individual position according to the above behavior”, and their complexities are O(N), O(N2), O(N), and O(N), respectively. Step “QPSO process”: The computational complexity of this step mainly includes • “Subpopulation-QPSO individuals were optimized by QPSO”, “Update the particle’s local t+1 best point Pi and global best point Pbest”, and “Update the position xij by Equation (10)”, and their complexities are all O(N). 2. Other step.

The computational complexity of Step “Initialization”, Step “Evaluation”, Step • “Combination”, and Step “Finding the best solution” are all O(N).

Therefore, in one generation AKQPSO, the worst-case complexity is O(N2).

6. Conclusions In this paper, aiming at the disadvantage of poor local search ability of KH algorithm, we optimized it by simulated annealing strategy and QPSO algorithm, and proposed a new algorithm: AKQPSO. This algorithm was compared with eight other excellent algorithms, and the computational accuracy of AKQPSO was tested by the 100-Digit Challenge problem. As predicted before the experiment, the computational accuracy of AKQPSO algorithm is the highest among these algorithms. Moreover, we also adjusted the parameters of AKQPSO through experiments to further improve the computational accuracy. By calculating the accuracy of the 100-Digit Challenge problem, our experiment provided a new method to study the accuracy of the algorithm, and the paper provided a very high accuracy algorithm. However, the research of this paper still has some limitations. The accuracy of the algorithm in 100-Digit Challenge problem is very high, but it is unclear whether AKQPSO still has high accuracy in general problems. In the future, we can focus on other aspects. Firstly, besides the simulated annealing strategy and QPSO algorithm, we could look for other metaheuristic algorithms [85] that can be used to improve the search ability of KH and carry out in-depth research such as bat algorithm (BA) [86,87], biogeography-based optimization (BBO) [84,88], cuckoo search (CS) [82,89–92], earthworm optimization algorithm (EWA) [93], elephant herding optimization (EHO) [94,95], moth search (MS) algorithm [96], firefly algorithm (FA) [97], artificial bee colony (ABC) [98–100], harmony search (HS) [101], monarch butterfly optimization (MBO) [102,103], and genetic programming (GP) [104], as well as more recent research. Secondly, for the parameters in the algorithm, besides the proportion of subpopulation, we can also study the influence of other parameters on the computational accuracy of AKQPSO. Mathematics 2020, 8, 1403 19 of 23

Thirdly, now we just use two kinds of algorithms to make them cooperate and optimize in a relatively simple way, and we can also study how to make them cooperate more eﬃciently through other methods. Finally, although the 100-Digit Challenge is a classical problem, it is still less used for testing computational accuracy of algorithms, and there is still a large development space in this area.

Author Contributions: Conceptualization, C.-L.W. and G.-G.W.; methodology, C.-L.W.; software, G.-G.W.; validation, C.-L.W.; formal analysis, G.-G.W.; investigation, C.-L.W.; resources, G.-G.W.; data curation, C.-L.W.; writing—original draft preparation, C.-L.W.; writing—review and editing, C.-L.W.; visualization, C.-L.W.; supervision, G.-G.W.; project administration, G.-G.W.; funding acquisition, G.-G.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Natural Science Foundation of China, grant number U1706218, 41576011, 41706010 and 61503165. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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